The cumulant lattice Boltzmann equation in three dimensions: Theory and validation

We propose, analyze, and validate a lattice Boltzmann model with a cumulant collision operator. The new model is analytically and numerically shown to poses smaller errors than a moment based Multiple Relaxation Time lattice Boltzmann model. We demonstrate the usability of the cumulant lattice Boltzmann model by simulations of flow around a sphere for Reynolds numbers from 200 to 105.

[1]  Bastien Chopard,et al.  Lattice Boltzmann method with regularized pre-collision distribution functions , 2006, Math. Comput. Simul..

[2]  Yassin A. Hassan,et al.  The effect of lattice models within the lattice Boltzmann method in the simulation of wall-bounded turbulent flows , 2013, J. Comput. Phys..

[3]  Pierre Lallemand,et al.  On rotational invariance of lattice Boltzmann schemes , 2014, Comput. Math. Appl..

[4]  Chengwen Zhong,et al.  Multiple-relaxation-time lattice Boltzmann method for study of two-lid-driven cavity flow solution multiplicity , 2014 .

[5]  Pietro Asinari,et al.  Asymptotic analysis of multiple-relaxation-time lattice Boltzmann schemes for mixture modeling , 2008, Comput. Math. Appl..

[6]  Shiyi Chen,et al.  A Lattice Boltzmann Subgrid Model for High Reynolds Number Flows , 1994, comp-gas/9401004.

[7]  M. Krafczyk,et al.  Turbulent jet computations based on MRT and Cascaded Lattice Boltzmann models , 2012, Comput. Math. Appl..

[8]  J. Korvink,et al.  Bubble functions for the lattice Boltzmann method and their application to grid refinement , 2009 .

[9]  J. Korvink,et al.  A factorized central moment lattice Boltzmann method , 2009 .

[10]  C. K. Chong,et al.  Rotational invariance in the three-dimensional lattice Boltzmann method is dependent on the choice of lattice , 2011, J. Comput. Phys..

[11]  M. Mcpherson,et al.  Introduction to fluid mechanics , 1997 .

[12]  Karl Heinz Hoffmann,et al.  The cumulant method applied to a mixture of Maxwell gases , 2002 .

[13]  P. Dellar.  Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results , 1993, Journal of Fluid Mechanics.

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  Martin Geier,et al.  Multi-thread implementations of the lattice Boltzmann method on non-uniform grids for CPUs and GPUs , 2011, Comput. Math. Appl..

[17]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation , 1993, Journal of Fluid Mechanics.

[18]  P. Moin,et al.  Eddies, streams, and convergence zones in turbulent flows , 1988 .

[19]  Dominique d'Humières,et al.  Viscosity independent numerical errors for Lattice Boltzmann models: From recurrence equations to "magic" collision numbers , 2009, Comput. Math. Appl..

[20]  Karl Heinz Hoffmann,et al.  The cumulant method for the space-homogeneous Boltzmann equation , 2005 .

[21]  Pierre Lallemand,et al.  Quartic parameters for acoustic applications of lattice Boltzmann scheme , 2010, Comput. Math. Appl..

[22]  L. Luo,et al.  Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation , 1997 .

[23]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[24]  Alexander N Gorban,et al.  Nonequilibrium entropy limiters in lattice Boltzmann methods , 2007, 0704.0043.

[25]  J. Korvink,et al.  Cascaded digital lattice Boltzmann automata for high Reynolds number flow. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Martin Geier,et al.  Lattice Boltzmann large eddy simulation of subcritical flows around a sphere on non-uniform grids , 2011, Comput. Math. Appl..

[27]  D. d'Humières,et al.  Thirteen-velocity three-dimensional lattice Boltzmann model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[29]  Steffen Seeger,et al.  The cumulant method for computational kinetic theory , 2000 .

[30]  Wolfgang Schröder,et al.  A lattice-Boltzmann method with hierarchically refined meshes , 2013 .

[31]  P. Asinari.  Generalized local equilibrium in the cascaded lattice Boltzmann method. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  M. Junk,et al.  Asymptotic analysis of the lattice Boltzmann equation , 2005 .

[33]  Dominique d'Humières,et al.  Multireflection boundary conditions for lattice Boltzmann models. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Sanjoy Banerjee,et al.  On the Three-Dimensional Central Moment Lattice Boltzmann Method , 2011, 1202.6081.

[35]  Jaber Almedeij,et al.  Drag coefficient of flow around a sphere: Matching asymptotically the wide trend , 2008 .

[36]  Ruddy Brionnaud,et al.  Solution to industry benchmark problems with the lattice-Boltzmann code XFlow , 2012 .

[37]  Rory A. Fisher,et al.  The Derivation of the Pattern Formulae of Two‐Way Partitions from those of Simpler Patterns , 1932 .

[38]  Viriato Semião,et al.  Truncation errors and the rotational invariance of three-dimensional lattice models in the lattice Boltzmann method , 2014, J. Comput. Phys..

[39]  R. Clift,et al.  Bubbles, Drops, and Particles , 1978 .

[40]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[41]  P. Lallemand,et al.  Momentum transfer of a Boltzmann-lattice fluid with boundaries , 2001 .

[42]  Mikhail D. Mikhailov,et al.  The drag coefficient of a sphere: An approximation using Shanks transform , 2013 .

[43]  Yan Peng,et al.  Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  François Dubois,et al.  Equivalent partial differential equations of a lattice Boltzmann scheme , 2008, Comput. Math. Appl..

[45]  D. d'Humières,et al.  Multiple–relaxation–time lattice Boltzmann models in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[46]  G. Taylor,et al.  Mechanism of the production of small eddies from large ones , 1937 .

[47]  Paul J. Dellar,et al.  Lattice Boltzmann algorithms without cubic defects in Galilean invariance on standard lattices , 2014, J. Comput. Phys..

[48]  Abbas Fakhari,et al.  Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  P. Lallemand,et al.  Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[50]  Wei Shyy,et al.  Force evaluation in the lattice Boltzmann method involving curved geometry. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Paul J. Dellar,et al.  Incompressible limits of lattice Boltzmann equations using multiple relaxation times , 2003 .

[52]  I. Ginzburg.  Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation , 2005 .